Flow Equations: Singular SPDEs and Quantum Field Theory

This lecture presents an overview of recent research on singular stochastic partial differential equations (SPDEs) and their applications in quantum field theory (QFT). The instructor discusses three main projects, starting with the flow equation approach to singular SPDEs, particularly focusing on the generalized parabolic Anderson model. The lecture explains how to regularize ill-posed equations and the conditions under which solutions exist. The second project involves constructing a measure for a critical fermionic QFT model using flow equations, emphasizing the importance of non-commutative properties of fermionic fields. The final project addresses the Euclidean invariance of a specific QFT model through stochastic quantization, detailing the challenges and methods used to prove invariance under transformations. The instructor concludes by outlining future research directions, including the study of correlation decay in QFT models and the exploration of dynamical Sine-Gordon models. Overall, the lecture highlights the intersection of probability theory, PDEs, and QFT, showcasing innovative approaches to complex mathematical problems.